Energy-efficient predictive control for trams incorporating disjunctive time constraints from traffic lights

Tram operations are often blocked by traffic lights, leading to frequent decelerations and re-accelerations that increase operational energy consumption. This paper focuses on tram energy-efficient control problem incorporating time constraints from traffic lights that have multiple feasible green time windows (GTWs). We formulate the problem as a mixed-integer nonlinear program (MINLP), where binary variables are assigned to model disjunctive time constraints of the GTWs. To address computational challenge of solving the MINLP, we reformulate it as a tractable nonlinear program (NLP). Specifically, an equivalent NLP is first presented by replacing the integrality constraint with nonlinear constraints, and then the nonlinear constraints are relaxed and penalized into cost functions. To recover a solution of the MINLP, we propose a computationally efficient sequential quadratic programming algorithm in a shrinking horizon model predictive control framework, which updates the penalty parameter and quadratic programming subproblems in parallel. The solution obtained from the subproblem is feasible in each iteration, and convergence of the feasibility iterations can be enforced by the updated penalty. The performance of the proposed approach is investigated on different scenarios using real-life tram data. Results show that the method is able to generate energy-efficient driving trajectories in a dynamic environment, while crossing traffic lights in effective GTWs without unnecessary decelerations and re-accelerations.